Moment generating function

A moment generating function is the function $M_X(t)=\mathbb{E}[e^{tX}]$ of a real parameter $t$ defined for a random variable $X$ on all values of $t$ for which the expectation is finite (often an interval containing $0$).

If $M_X(t)$ is finite on an open interval around $0$, then $M_X^{(k)}(0)=\mathbb{E}[X^k]$, linking it directly to moments . The closely related cumulant generating function is $\log M_X(t)$ (when defined), and the characteristic function can be used when the mgf does not exist.

Examples:

• If $X\sim N(\mu,\sigma^2)$, then $M_X(t)=\exp\!\left(\mu t+\tfrac{1}{2}\sigma^2 t^2\right)$ for all real $t$.
• If $X\sim\mathrm{Bernoulli}(p)$, then $M_X(t)=(1-p)+p\,e^{t}$ for all real $t$.
• If $X\sim\mathrm{Exp}(\lambda)$ with $\lambda>0$, then $M_X(t)=\frac{\lambda}{\lambda-t}$ for $t<\lambda$.